1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Romashka-Z-Leto [24]
3 years ago
15

Find the locus of a point such that the sum of its distance from the point ( 0 , 2 ) and ( 0 , -2 ) is 6.

Mathematics
2 answers:
jok3333 [9.3K]3 years ago
5 0

Answer:

\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1

Step-by-step explanation:

We want to find the locus of a point such that the sum of the distance from any point P on the locus to (0, 2) and (0, -2) is 6.

First, we will need the distance formula, given by:

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Let the point on the locus be P(x, y).

So, the distance from P to (0, 2) will be:

\begin{aligned} d_1&=\sqrt{(x-0)^2+(y-2)^2}\\\\ &=\sqrt{x^2+(y-2)^2}\end{aligned}

And, the distance from P to (0, -2) will be:

\displaystyle \begin{aligned} d_2&=\sqrt{(x-0)^2+(y-(-2))^2}\\\\ &=\sqrt{x^2+(y+2)^2}\end{aligned}

So sum of the two distances must be 6. Therefore:

d_1+d_2=6

Now, by substitution:

(\sqrt{x^2+(y-2)^2})+(\sqrt{x^2+(y+2)^2})=6

Simplify. We can subtract the second term from the left:

\sqrt{x^2+(y-2)^2}=6-\sqrt{x^2+(y+2)^2}

Square both sides:

(x^2+(y-2)^2)=36-12\sqrt{x^2+(y+2)^2}+(x^2+(y+2)^2)

We can cancel the x² terms and continue squaring:

y^2-4y+4=36-12\sqrt{x^2+(y+2)^2}+y^2+4y+4

We can cancel the y² and 4 from both sides. We can also subtract 4y from both sides. This leaves us with:

-8y=36-12\sqrt{x^2+(y+2)^2}

We can divide both sides by -4:

2y=-9+3\sqrt{x^2+(y+2)^2}

Adding 9 to both sides yields:

2y+9=3\sqrt{x^2+(y+2)^2}

And, we will square both sides one final time.

4y^2+36y+81=9(x^2+(y^2+4y+4))

Distribute:

4y^2+36y+81=9x^2+9y^2+36y+36

The 36y will cancel. So:

4y^2+81=9x^2+9y^2+36

Subtracting 4y² and 36 from both sides yields:

9x^2+5y^2=45

And dividing both sides by 45 produces:

\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1

Therefore, the equation for the locus of a point such that the sum of its distance to (0, 2) and (0, -2) is 6 is given by a vertical ellipse with a major axis length of 3 and a minor axis length of √5, centered on the origin.

LUCKY_DIMON [66]3 years ago
4 0

ANSWER:

We need to know a distance formula to find the locus.

By distance formula,

  • √[x² + (y - 2)²] + √[x² + (y + 2)²] = 6.
  • √(x² + y² - 4y + 4) + √(x² + y² + 4y + 4) = 6.

Let x² + y² + 4 = a.

  • √(a - 4y) + √(a + 4y) = 6.
  • √(a - 4y) = 6 - √(a + 4y).

Squaring both sides,

  • a - 4y = a + 4y + 36 - 12 × √(a + 4y).
  • 8y + 36 = 12 × √(a + 4y).
  • 2y + 9 = 3 × √(a + 4y).

Squaring again,

  • 4y² + 36y + 81 = 9a + 36y.
  • 9a - 4y² - 81 = 0.

Putting back the original value of a,

  • 9x² + 5y² - 45 = 0.

You might be interested in
Okay I don't understand how do I do this. It say :
Lilit [14]

2x+3=17

subtract 3 from each side

2x+3 -3 = 17 -3

2x = 14

divide by 2 on each side

2x/2 = 14/2

x = 7

6 0
3 years ago
Read 2 more answers
The snack bar at your school has added sushi to its menu. The ingredients for one roll include sushi​ rice, seaweed​ sheets, cuc
amid [387]

Answer:

The answer si:  

- 16 servings - 6 oz of salmon

- 24 servings - 9 oz of salmon

- 80 servings - 30 oz of salmon

- 100 servings - 37.5 oz of salmon

1 roll is 8 servings and it is also made of 3oz of smoked salmon. That means that 3oz of smoked salmon is needed fo 8 servings.

Now, let's made some proportions:

16 servings:

3 oz is for 8 servings, how much oz is for 16 servings:

3 : 8 = x : 16

x = 3 · 16 ÷ 8 = 6 oz

24 servings:

3 oz is for 8 servings, how much oz is for 24 servings:

3 : 8 = x : 24

x = 3 · 24 ÷ 8 = 9 oz

80 servings:

3 oz is for 8 servings, how much oz is for 80 servings:

3 : 8 = x : 80

x = 3 · 80 ÷ 8 = 30 oz

100 servings:

3 oz is for 8 servings, how much oz is for 100 servings:

3 : 8 = x : 100

x = 3 · 100 ÷ 8 = 37.5 oz

5 0
3 years ago
Plz help for brianly
ozzi

Answer:

190.

Step-by-step explanation:

4 0
3 years ago
Write the standard equation of the circle with the given center that passes through the given point
Kitty [74]
Circle equation
(x - h)^2 + (y - k)^2 = r^2
Since radius isn't given use the center and point, with the distance formula, to find the radius.
D=sqrt((x2-x2)^2+)y2-y1)^2)
D=sqrt((1-0)^2+(2-6)^2
D=sqrt(1+16)
D=sqrt 17
radius = sqrt 17
(h, k) is the center and r is the radius
(x - 1)^2 + (y - 2)^2 = 17
5 0
3 years ago
Please help. I don't understand this.​
klio [65]

Answer:

radius=6

Step-by-step explanation:

arc ACB=6xPi

Pi=3.14159

arc ACB=18.85 (rounded to 2 places)

ACB is a half circle so a full circle would be 18.85x2 or 37.7

circumference (whole circle)=2 x Pi x radius

37.7=2 x 3.14159 x radius

radius= 37.7/(2 x 3.14159)

radius=6

7 0
3 years ago
Other questions:
  • Suppose you are given the task of constructing a soup can that holds 400mL of water (1 mL = 1 cm3 ). The material for the base a
    8·1 answer
  • Find each product by factoring the tens.
    13·1 answer
  • Classify abc by its angles if m
    7·1 answer
  • If the parent function y= csc x goes through a transformation to produce 1+csc (x-π), what is the resultant shift in the functio
    10·1 answer
  • 9x+9+5? how do you solve this.
    8·1 answer
  • X^2+6x+4 equivalent
    12·1 answer
  • What is the measure of the missing angle? Please answer fast!
    7·2 answers
  • Jana paid $48 for 4 tickets to a movie. How much would it cost for 8 tickets? I'm aware you could probably do this problem witho
    13·2 answers
  • What is the perimeter of this<br> polygon?<br> A_3 cm<br> 9 cm<br> C<br> [?] cm<br> B 3 cm
    7·1 answer
  • Can someone help me please
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!